Vector Bundle Valued Differential Forms on $\mathbb{N} Q$-manifolds

TL;DR

This paper characterizes vector bundle valued differential forms on non-negatively graded $ Q$-manifolds, unifying existing results and introducing new insights into their geometric structures like symplectic, contact, and higher order forms.

Contribution

It provides a unified description of vector bundle valued forms on $ Q$-manifolds and presents new proofs and results for various geometric structures in this context.

Findings

Unified description of differential forms on $ Q$-manifolds.

New results on symplectic, contact, and involutive structures.

Extension to higher order forms like presymplectic and multisymplectic structures.

Abstract

Geometric structures on $\mathbb N Q$-manifolds, i.e.~non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contact structures and, more generally, distributions are in this class. We describe vector bundle valued differential forms on non-negatively graded manifolds in terms of non-graded geometric data. Moreover, we use this description to present, in a unified way, novel proofs of known results, and new results about degree one $\mathbb N Q$-manifolds equipped with certain geometric structures, namely symplectic structures, contact structures, involutive distributions (already present in literature) and locally conformal symplectic structures, and generic vector bundle valued higher order forms, in particular presymplectic and multisymplectic structures (not yet present in literature).